Dynamic Programming Principle for Backward Doubly Stochastic Recursive Optimal Control Problem and Sobolev Weak Solution of The Stochastic Hamilton-Bellman Equation

TL;DR

This paper establishes a dynamic programming principle for backward doubly stochastic recursive optimal control problems and proves that the value function uniquely solves the associated stochastic Hamilton-Jacobi-Bellman equation in a Sobolev weak sense.

Contribution

It introduces a novel dynamic programming framework for backward doubly stochastic control problems and demonstrates the Sobolev weak solution property of the value function.

Findings

The value function satisfies the stochastic Hamilton-Jacobi-Bellman equation.

The paper proves the uniqueness of the Sobolev weak solution.

A new approach to backward doubly stochastic control problems is developed.

Abstract

In this paper, we study backward doubly stochastic recursive optimal control problem where the cost function is described by the solution of a backward doubly stochastic differential equation. We give the dynamical programming principle for this kind of optimal control problem and show that the value function is the unique Sobolev weak solution for the corresponding stochastic Hamilton-Jacobi-Bellman equation.